Spectral Duality for Planar Billiards

For a bounded open domain $\Omega$ with connected complement in ${\bf R}^2$ and piecewise smooth boundary, we consider the Dirichlet Laplacian $-\Delta_\Omega$ on $\Omega$ and the S-matrix on the complement $\Omega^c$. We show that the on-shell S-matrices ${\bf S}_k$ have eigenvalues converging to 1 as $k\uparrow k_0$ exactly when $-\Delta_\Omega$ has an eigenvalue at energy $k_0^2$. This includes multiplicities, and proves a weak form of ``transparency'' at $k=k_0$. We also show that stronger forms of transparency, such as ${\bf S}_{k_0}$ having an eigenvalue 1 are not expected to hold in general.Comment: 33 pages, Postscript, A

Publisher Correction: Variations in Dysbindin-1 are associated with cognitive response to antipsychotic drug treatment.

In the original version of this Article, references in the Methods section incorrectly referred to references in the Supplementary References section. The relevant references (now numbered 20, 27, 42, 47, 69-80) have been removed from the Supplementary References section of the Supplementary Information file and added to the References section of the main manuscript, in both the PDF and HTML versions of the Article

Identification of common genetic risk variants for autism spectrum disorder

Autism spectrum disorder (ASD) is a highly heritable and heterogeneous group of neurodevelopmental phenotypes diagnosed in more than 1% of children. Common genetic variants contribute substantially to ASD susceptibility, but to date no individual variants have been robustly associated with ASD. With a marked sample-size increase from a unique Danish population resource, we report a genome-wide association meta-analysis of 18,381 individuals with ASD and 27,969 controls that identified five genome-wide-significant loci. Leveraging GWAS results from three phenotypes with significantly overlapping genetic architectures (schizophrenia, major depression, and educational attainment), we identified seven additional loci shared with other traits at equally strict significance levels. Dissecting the polygenic architecture, we found both quantitative and qualitative polygenic heterogeneity across ASD subtypes. These results highlight biological insights, particularly relating to neuronal function and corticogenesis, and establish that GWAS performed at scale will be much more productive in the near term in ASD.Peer reviewe

Integral representations of solutions of the Helmholtz equation with application to diffraction by a strip

Electrical Engineering, Mathematics and Computer Scienc

The long and short of It: long-chain fatty acids and long-term outcomes for premature infants

Maria Makrides, Ronald E. Kleinma